A Preference Programming Approach to Make the Even Swaps Method Even Easier

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A Preference Programming Approach to Make the
Even Swaps Method Even Easier

• Jyri Mustajoki
• Raimo P. Hämäläinen
• Systems Analysis Laboratory
• Helsinki University of Technology
• www.sal.hut.fi

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Outline

• The Even Swaps method
• Hammond, Keeney and Raiffa (1998, 1999)
• A new combined Even Swaps / Preference
  Programming approach
• PAIRS method (Salo and Hämäläinen, 1992)
• Additive MAVT model of the problem
• Intervals to model incomplete information
• Support for different phases of the Even Swaps
  process
• Smart-Swaps Web software
• The first software for supporting the method

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Even Swaps

• Multicriteria method to find the best alternative
• An even swap
• A value trade-off, where a consequence change in
  one attribute is compensated with a comparable
  change in some other attribute
• A new alternative with these revised consequences
  is equally preferred to the initial one
• ? The new alternative can be used instead

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Elimination process

• Carry out even swaps that make
• Alternatives dominated (attribute-wise)
• There is another alternative, which is equal or
  better than this in every attribute, and better
  at least in one attribute
• Attributes irrelevant
• Each alternative has the same value on this
  attribute
• ? These can be eliminated
• Process continues until one alternative, i.e.
  the best one, remains

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Practical dominance

• If alternative y is slightly better than
  alternative x in one attribute, but worse in all
  or many other attributes
• ? x practically dominates y
• ? y can be eliminated
• Aim to reduce the size of the problem in obvious
  cases
• Eliminate unnecessary even swap tasks

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Example

• Office selection problem (Hammond et al. 1999)

An even swap
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Supporting Even Swaps with Preference Programming

• Even Swaps process carried out as usual
• The DMs preferences simultaneously modeled with
  Preference Programming
• Intervals allow us to deal with incomplete
  information about the DMs preferences
• Trade-off information given in the even swaps can
  be used to update the model
• ? Suggestions for the Even Swaps process
• Generality of assumptions of Even Swaps preserved

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Supporting Even Swaps with Preference Programming

• Support for
• Identifying practical dominances
• Finding candidates for the next even swap
• Both tasks need comprehensive technical screening
• Idea supporting the process not automating it

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Decision support
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Assumptions in the Preference Programming model

• Additive value function
• Not a very restrictive assumption
• Weight ratios and component value functions are
  initially within some reasonable bounds
• General bounds for these often assumed
• E.g. practical dominance implicitly assumes
  reasonable bounds for the weight ratios

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Preference Programming The PAIRS method

• Imprecise statements with intervals on
• Attribute weight ratios (e.g. 1/5 ? w1 / w2 ? 5)
• ? Feasible region for the weights
• Alternatives ratings (e.g. 0.6 ? v1(x1) ? 0.8)
• ? Intervals for the overall values
• Lower bound for the overall value of x
• Upper bound correspondingly

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Initial assumptions produce bounds

• For the weight ratios
• For the ratings
• Modeled with exponential value functions
• Any monotone value functions within the bounds
  allowed
• Additional bounds for the
  min/max slope

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Use of trade-off information

• With each even swap the user reveals new
  information about her preferences
• This trade-off information can be utilized in the
  process
• ? Tighter bounds for the weight ratios obtained
  from the given even swaps
• ? Better estimates for the values of the
  alternatives

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Practical dominance

• An alternative which is practically dominated
  cannot be made non-dominated with any reasonable
  even swaps
• Analogous to pairwise dominance concept in
  Preference Programming

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Pairwise dominance

• x dominates y in a pairwise sense if
• i.e. if the overall value of x is greater than
  the one of y with any feasible weights of
  attributes and ratings of alternatives
• ? Any pairwisely dominated alternative can be
  considered to be practically dominated

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Candidates for even swaps

• Aim to make as few swaps as possible
• Often there are several candidates for an even
  swap
• In an even swap, the ranking of the alternatives
  may change in the compensating attribute
• ? One cannot be sure that the other alternative
  becomes dominated with a certain swap

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Applicability index

• Assume y is better than x only in attribute i
• Applicability index of an even swap, where a
  change xi?yi is compensated in attribute j, to
  make y dominated
• Indicates how close to making y dominated we can
  get with this swap
• The bigger d is, the more likely it is to reach
  dominance

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Applicability index

• Ratio between
• The minimum feasible rating change in the
  compensating attribute to reach dominance and
• The maximum possible rating change that could be
  made in this attribute
• Worst case value for d
• Bounds include all the possible impecision
• Average case value for d
• Rating differences from linear value functions
• Weight ratios as averages of their bounds

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Example
Initial Range 85 - 50 A - C 950 - 500 1500 -1900
36 different options to carry out an even swap
that may lead to dominance E.g. change in Monthly
Cost of Montana from 1900 to 1500 Compensation
in Client Access d(M?B, Cost, Access)
((85-78)/(85-50)) / ((1900-1500)/(1900-1500))
0.20 d(M?L, Cost, Access) ((85-80)/(85-50))
/ ((1900-1500)/(1900-1500)) 0.14 Compensation
in Office Size d(M?B, Cost, Size)
((950-500)/(950-500)) / ((1900-1500)/(1900-1500))
1.00 d(M?L, Cost, Size) ((950-700)/(950-500
)) / ((1900-1500)/(1900-1500)) 0.56

(Average case values for d used)
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Comparison with MAVT
Even Swaps MAVT
Assumptions about the value function Not needed Needed Additive functions typically used
Elicitation burden No. of elicitations may become high Not known in advance Increases with the no. of alternatives Weight elicitation At least n-1 preference statements Value functions One for each attribute
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Comparison with MAVT
Even Swaps MAVT
Analysis of the results Dominance relations No relative scores Outcomes of the alternatives change during the process Overall scores for the alternatives Clear to interpret
Suitability Personal decision making Proposed approach makes the process easier Group and policy decisions Transparency of the process
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Smart-Swaps softwarewww.smart-swaps.hut.fi

• Identification of practical dominances
• Suggestions for the next even swap to be made
• Additional support
• Information about what can be achieved with
  each swap
• Notification of dominances
• Rankings indicated by colors
• Process history allows backtracking

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Problem definition
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Entering trade-offs
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Process history
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www.Decisionarium.hut.fi

• Software for different types of problems
• Smart-Swaps (www.smart-swaps.hut.fi)
• Opinions-Online (www.opinions.hut.fi)
• Global participation, voting, surveys group
  decisions
• Web-HIPRE (www.hipre.hut.fi)
• Value tree based decision analysis and support
• Joint Gains (www.jointgains.hut.fi)
• Multi-party negotiation support
• RICH Decisions (www.rich.hut.fi)
• Rank inclusion in criteria hierarchies

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Conclusions

• Modeling of the DMs preferences in Even Swaps
  with Preference Programming allows to
• Identify practical dominances
• Find candidates for even swaps
• Makes the Even Swaps process even easier
• Support provided as suggestions by the
  Smart-Swaps software

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References

• Hämäläinen, R.P., 2003. Decisionarium - Aiding
  Decisions, Negotiating and Collecting Opinions on
  the Web, Journal of Multi-Criteria Decision
  Analysis, 12(2-3), 101-110.
• Hammond, J.S., Keeney, R.L., Raiffa, H., 1998.
  Even swaps A rational method for making
  trade-offs, Harvard Business Review, 76(2),
  137-149.
• Hammond, J.S., Keeney, R.L., Raiffa, H., 1999.
  Smart choices. A practical guide to making better
  decisions, Harvard Business School Press, Boston.
• Mustajoki, J., Hämäläinen, R.P., 2005. A
  Preference Programming Approach to Make the Even
  Swaps Method Even Easier, Decision Analysis,
  2(2), 110-123.
• Salo, A., Hämäläinen, R.P., 1992. Preference
  assessment by imprecise ratio statements,
  Operations Research, 40(6), 1053-1061.
• Applications of Even Swaps
• Gregory, R., Wellman, K., 2001. Bringing
  stakeholder values into environmental policy
  choices a community-based estuary case study,
  Ecological Economics, 39, 37-52.
• Kajanus, M., Ahola, J., Kurttila, M., Pesonen,
  M., 2001. Application of even swaps for strategy
  selection in a rural enterprise, Management
  Decision, 39(5), 394-402.